### Home > APCALC > Chapter 12 > Lesson 12.1.3 > Problem12-33

12-33.

Write an equation for $\frac { d y } { d x }$ given each equation below. (Hint for part (c): Use natural log to rewrite this equation.)

1. $y = \int _ { 3 } ^ { x } \sqrt { 1 + e ^ { u } } d u$

$\frac{d}{dx}y=\frac{d}{dx}\int_3^x\sqrt{1+e^u}du$

$\frac{dy}{dx}=\sqrt{1+e^x}$

1. $\left\{ \begin{array} { l } { x ( t ) = \operatorname { cos } ( 2 t ) } \\ { y ( t ) = \operatorname { cos } ^ { - 1 } ( 2 t ) } \end{array} \right.$

$\frac{dy}{dx}=\frac{y^\prime(t)}{x^\prime(t)}$

1. $y = x^{x}$

$\ln(y)=x\ln(x)$

$\frac{1}{y}y^\prime=\ln(x)+1$

$y^\prime=y(\ln(x)+1)$

$y^\prime=x^x(\ln(x)+1)$

1. $y = x \csc(\ln(x))$

$y^\prime=\csc(\ln(x))+x(-\csc(\ln(x))\cot(\ln(x))\cdot\frac{1}{x}$